Graded Lie algebras of maximal class. (English) Zbl 0895.17031

An algebra of maximal class is a graded Lie algebra \(L= \bigoplus_{i=1}^\infty L_i\) over a field \(F\), where \(\dim(L_1)= 2\), \(\dim (L_1)\leq 1\) for \(i\geq 2\), and \([L_i, L_j]=: L_{i+1}\) for all \(i\geq 1\). For \(F\) of characteristic \(p>0\) the authors construct \(| F|^{\aleph_0}\) pairwise nonisomorphic nonsolvable algebras of maximal class and \(\max\{| F|, \aleph_0\}\) solvable ones. Both numbers are shown to be best possible. The construction given in this paper yields an extension of a tensor product of a maximal ideal of \(L\) and an extension of \(F\). The isomorphism type of \(L\) is determined by its sequence of two-step centralizers.


17B70 Graded Lie (super)algebras
17B50 Modular Lie (super)algebras
17B30 Solvable, nilpotent (super)algebras


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